Is Mega Millions’ $640 Million Jackpot a Good Bet?

The Mega Millions jackpot is now over half a billion dollars ($640 million, to be exact). So is it time to buy a lottery ticket?

Bloomberg News

No, or at least not according to an analysis performed by computer scientist Jeremy Elson last year, when the jackpot was a “mere” $380 million.

It’s actually not a crazy question. Sure, the lottery is a sucker’s bet in general. But the odds of winning, per the official Mega Millions web site are about 1 in 176 million. A ticket only costs a dollar. So anytime the jackpot is worth more than $176 million, a $1 ticket is a good bet, right?

Not so fast. First, that eye-popping jackpot figure doesn’t mean the winner gets an immediate check for $640 million. It’s paid over 26 years as an annuity. You can take a lump sum now, but you only get $462 million. Taxes bite off another 40% or so, depending on what state you live in. Still, the jackpot should still be worth around $275 million in cold, hard, after-tax cash. That’s still a lot more than $176 million. So the odds are looking — in true Hunger Games fashion — ever in your favor.

The problem, though, is that even if you win, there’s no guarantee you’d be the only winner. According to economist Stephen Bronars , lottery officials figure there’s about a 76% chance of someone picking a winner in tomorrow’s drawing, and that if there’s at least one winner, there will likely be two (or, more precisely, 1.9, on average). That would cut your likely winnings in half. Mr. Bronars actually still figures it’s worth the gamble, but he doesn’t appear to be factoring in taxes.

Still, let’s say no one picks a winner tomorrow, and the jackpot keeps growing. At some point, it’d get big enough to be worth the bet, right?

Don’t count on it. Mr. Elson notes that when jackpots go up, more people buy tickets, hoping for a shot at lottery riches. In fact, they seem to do so super-linearly — that is, the buying frenzy picks up pace as the jackpot grows. The more people who buy tickets, the greater the risk of multiple winners, and thus the lower the value of any payout. (Mr. Elson notes that there haven’t been many cases of truly super-sized jackpots, so it’s possible the sales-to-jackpot relationship breaks down at the upper reaches.)

Taking all this into account, Mr. Elson plots the expected value of a ticket against the size of the jackpot, and finds that it looks more or less like a bell-curve. That is, up to a certain point, higher jackpots mean higher expected payouts. But after a certain point, the risk of multiple winners overwhelms the higher jackpots, and the expected payout goes down. By Mr. Elson’s calculations, the best time to buy a Mega Millions ticket is when the jackpot is $420 million. Even then, though, the expected value of a ticket is just 69 cents. So you’d still be better off keeping your dollar.

But hey, you never know.

UPDATE: Numbers in this post were updated when the jackpot size was raised.

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